Guo-Fang Shi(石國芳)

School of Science,Xi’an University of Posts and Telecommunications,Xi’an 710061,China

Keywords: quantum dialogue,entanglement state,Bell measurement

1. Introduction

Quantum communication is a new way to communicate by using the principles of quantum mechanics. Its security does not depend on the computational complexity in mathematics, but is guaranteed by the physical principles, and is absolutely secure. Quantum communication mainly includes quantum key distribution(QKD),quantum secure direct communication(QSDC),quantum secret sharing(QSS),etc.

The task of QKD[1–3]is that it enables legitimate communicators to establish and share a secret key to encrypt and decrypt messages. Since the first QKD protocol based on a single-photon system was proposed by Bennett and Brassard in 1984(BB84 protocol),[1]many excellent protocols[2,3]have been put forward successively. Although QKD has the advantage of unconditional security in theory, it still provides potential opportunities for some eavesdropping behaviors[4–8]due to the imperfections of various devices in practical application. Among these imperfections, the defect of measurement equipment is particularly prominent. The solution to this problem is to build a nearly perfect device,which is obviously impractical. Fortunately,in 2012,Lo,Curty and Qi,three researchers,proposed a measurement-device-independent QKD(MID QKD) protocol,[9]which is independent of measurement devices. This QKD system is very different from the previous QKD scheme; that is, all the photons are measured by an untrusted third party. Such a system can be immune to all malicious attacks associated with measurement devices,and it will play a vital role in the development of QKD under real conditions in the future. Moreover,since the third party is in the middle of the two communication parties in the scheme,it can effectively extend the transmission distance of the QKD.All these remarkable advantages make this MID QKD one of the best candidates among all kinds of QKD protocols. It attracted the attention of many researchers as soon as it was reported, and has been an important research topic in recent years.[10–12]

QSDC is another important branch of quantum communication developed in the past 20 years.[13–21]It is a unique technology, which can transmit confidential information directly without the help of secret key and ciphertext. Early QSDC was a one-way transmission of secret information, in 2004, Nguyen first generalized it to a two-way situation, socalled quantum dialogue.[22]This is a very interesting concept,in which Alice and Bob can send secret information each other. Since then,quantum dialogue has gradually grown and attracted lots of attention. However, in 2008, Gaoet al.[23]and Tanet al.[24]independently demonstrated that the quantum dialogue protocol[22]has the drawback of “secret information leakage” from the perspective of information theory,which does not meet the requirements of secure communication. Any secure bidirectional quantum communication protocol or quantum dialogue scheme should not only resist the conventional attacks of eavesdropper,but also avoid the defect of“secret information leakage”.[25–27]

Recently,Niuet al.[28]and Zhouet al.[29]have presented the MID QSDC protocols. Inspired by their ideas,in this paper,we will design a MID quantum dialogue. In this protocol,Alice prepares N identical entangled EPR pairs as quantum channels, Bob prepares entangled EPR pairs according to his own secret information,and both communication parties send their photons sequences to the third party Charlie by means of the quantum data block transmission.[14–16]After a series of security tests, if the channel is secure, Alice will perform the Bell measurement and publish the results. The secret of Bob is transmitted to Alice by means of teleportation, and the secret of Alice is decoded by Bob with the knowledge of dense coding, thus, quantum dialogue is realized successfully. The security analysis shows that this scheme can not only effectively resist common attacks by eavesdroppers,but also avoid the defect of “secret information leakage”. Due to the use of the technology of MID, it can also be immune to all security loopholes associated with the measurement device. Compared with the previous MID quantum dialogue,[27]it is more concise and has higher communication efficiency. Besides, it is feasible with present-day technology.

2. Measurement-device-independent quantum dialogue scheme

An EPR pair is a commonly used pair of entangled photons in quantum communication, and its state is one of the following:

Here 0 and 1 represent the horizontal and vertical polarization states of the photon. In this scheme,two communication parties agree that Bob prepares different quantum states according to the classic secret information bit 0 or 1 he wants to send. The specific corresponding relationship is

The specific scheme consists of the following seven steps, as shown in Fig.1.

Fig. 1. The illustration of MID quantum dialogue scheme. The black photon pairs (a1,t1)(a′1,t′1)(a2,t2)(a′2,t′2)···(aN,tN)(a′N,t′N) at Alice’s side are the EPR state |?+〉at in Eq. (1). The black photon pairs(b1,b′1),(b2,b′2)···(bN,b′N)prepared by Bob are one of EPR states according to Eqs.(1)and(2). The grey single-photon qubits are randomly prepared in one of the BB84 states

Table 1.The source of photon pairs(ti,bi)and the use of Bell measurements in every situation.

In Table 1,two photons in case(1)are both from the entangled pair of Alice and Bob,and Charlie’s Bell measurement is equivalent to allowing the photons in different entangled pairs to achieve entanglement swapping. Such photon pairs are most cases,and will be used to realize quantum dialogue.The photon pairs in case(3)will be discarded because only a small number of single photons are inserted. In order to make the security test effective,Alice and Bob only use the photons in case(2)to analyze the security of the step 2. The details are as follows:

Alice and Bob exchange the basis information and quantum state of single photons through classical communication.For those single photons with different preparation basis,Bell measurement will lead to any one of four Bell states,as shown in Eqs.(4)and(5),it is not useful for security check and analysis.The decomposition of two photons with identical basis in terms of Bell state are shown as Eqs.(6)and(7),and there are only two possible Bell states. Using these single photons with the same basis,Alice and Bob can estimate that the transmission in Step 2 is secure or not. Such a security check is completely equivalent to the literature,[9]and can also be regarded as the“time reversal”of the QKD protocol(BBM92).[3]The behavior of the eavesdropper Eve would cause some errors when he chooses the wrong measurement basis. Alice and Bob compare and analyze, if the error rate is higher than a threshold, it indicates that Step 2 is insecure, and both parties abandon the above transmission and start again. If they confirm that the quantum channel is safe,then the next step is started.

Table 2. Suppose Bob’s third secret is 1 and he prepares initial state|ψ?〉b3b′3.UT is a unitary operation to realize the teleportation of(b3,b′3)to(a3,a′3). The correspondence between Bell measurement outcomes(t3,b3),(t′3,b′3)of Charlie and UT performed by Alice.

For Alice, from the Bell measurement result(UMUTaj,a′j) andUM, she can infer the initial state (bj,b′j)prepared by Bob and extract Bob’s 1-bit classical secret information. For Bob,according to the Bell measurement result of(UMUTaj,a′j) and his initial state, he can also extract Alice’s 1-bit secret message. The quantum dialogue has been realized. In the MID quantum communication scheme,whether it is based on a single-photon state[9,26,27]or an entangled state as a quantum channel,[28]the Bell measurement equipment has always been an indispensable component. Bell measurement using linear optical elements and single-photon detectors is favored because of its simple structure. Such Bell measurement devices can only distinguish two of the four Bell states,that is, they can only identify Bell states probabilistically.This fact will result in the halving of the efficiency both in the security check and the decoding message process,though it does not affect the security of the protocol. It should be noted that four EPR states must be distinguished completely in the current MID quantum dialogue protocol because the secret information in Step 7 carried by Bell state is 2 bits.That is to say, non-linear optical elements[30–32]should be used to accomplish complete Bell measurement in the present scheme.

3. Security analysis of the protocol

Now we will prove the security of this protocol.

4. Discussion and conclusion

Quantum dialogue belongs to the category of QSDC,in the current MID quantum dialogue scheme, it mainly involves Bell state preparation,single photon preparation,quantum storage, Bell state measurement and other technologies,and these technologies have been mature in the experimental realization of QSDC. In 2017, academician Guo guangcan’s research group realized an entanglement-based QSDC with the help of atom ensemble quantum memory.[34]In the same year, Zhuet al.increased the transmission distance of the entanglement-based QSDC in the optical fiber in the order of kilometers,[35]and achieved long-distance safe communication. In 2020, academician Pan Jianwei and his collaborators realized MID QKD in free space for the first time.[36]All these experimental successes undoubtedly show that the MID quantum dialogue protocol is achievable with current technology.